It is well known that mechanical disturbances can be used to establish acoustic waves in earth formations surrounding a borehole, and the properties of these waves can be measured to obtain important information about the formations through which the waves have propagated. Parameters of compressional, shear, and Stoneley waves, such as their velocity (or its reciprocal, slowness) in the formation and in the borehole, can be indicators of formation characteristics that help in evaluation of the location and/or producibility of hydrocarbon resources.
An example of a logging device that has been used to obtain and analyze sonic logging measurements of formations surrounding an earth borehole is called a Dipole Shear Sonic Imager (“DSI”—trademark of Schlumberger), and is of the general type described in Harrison et al., “Acquisition and Analysis of Sonic Waveforms From a Borehole Monopole And Dipole Source For The Determination Of Compressional And Shear Speeds And Their Relation To Rock Mechanical Properties And Surface Seismic Data”, Society of Petroleum Engineers, SPE 20557, 1990.
An acoustic source in a fluid-filled borehole generates headwaves as well as relatively stronger borehole-guided modes. A standard sonic measurement system consists of placing a piezoelectric or electrodynamic source and an array of hydrophone receivers inside a fluid-filled borehole. The acoustic source is configured in the form of either a monopole or a dipole source. The source bandwidth typically ranges from 0.5 to 20 kHz. A monopole source generates primarily the lowest-order axisymmetric mode, also referred to as the Stoneley mode, together with compressional and shear headwaves. In contrast, a dipole source primarily excites the lowest-order borehole flexural mode together with compressional and shear headwaves. The headwaves are caused by the coupling of the transmitter acoustic energy to plane waves in the formation that propagate along the borehole axis. An incident compressional wave in the borehole fluid produces critically refracted compressional waves in the formation. These refracted waves along the borehole surface are known as compressional headwaves. The critical incidence angle θi=sin−1(Vf/Vc), where Vf is the compressional wave speed in the borehole fluid; and Vc is the compressional wave speed in the formation. As the compressional headwave travels along the interface, it radiates energy back into the fluid that can be detected by hydrophone receivers placed in the fluid-filled borehole. In fast formations, the shear headwave can be similarly excited by a compressional wave at the critical incidence angle θi=sin−1(Vf/Vc), where Vf is the shear wave speed in the formation. In a homogeneous and isotropic model of fast formations, compressional and shear headwaves can be generated by a monopole source placed in a fluid-filled borehole for determining the formation compressional and shear wave speed (Kimball and Marzetta, 1984). It is known that refracted shear headwaves cannot be detected in slow formations (where the shear wave velocity is less than the borehole-fluid compressional velocity) with receivers placed in the borehole fluid. In slow formations, formation shear velocities are obtained from the low-frequency asymptote of borehole flexural dispersion. Standard processing techniques exist for the estimation of formation shear velocities in either fast or slow formations from an array recorded dipole waveforms. The radial depth of investigation in the case of headwave logging is dependent on the transmitter to receiver spacing and any in-situ radial variations in formation properties that might be present. The radial depth of investigation in the case of modal logging is well characterized and it extends to about a wavelength. This implies that the low- and high-frequencies probe deep and shallow into the formation, respectively. Existing sonic tools, such as the above-referenced “DSI” are capable of providing the far-field compressional, fast-shear, and slow-shear velocities away from the near-wellbore alterations.
The effective formation volume probed by borehole sonic waves is generally, subject to (1) near-wellbore stress distributions caused by the far-field formation stresses; (2) wellbore mud pressure; and (3) pore pressures. Formation stresses play an important role in geophysical prospecting and developments of oil and gas reservoirs. Both the direction and magnitude of these stresses are required in (a) planning for borehole stability during directional drilling, (b) hydraulic fracturing for enhanced production, and (c) selective perforation for prevention of sanding during production. A detailed knowledge of formation stresses also helps in management of reservoirs that are prone to subsidence caused by a significant reduction in pore pressure and an associated increase in the effective stress that exeeds the in-situ rock strength. In addition, the magnitude and orientation of the in-situ stresses in a given field have a significant influence on permeability distribution that help in a proper planning of wellbore trajectories and injection schemes for water or steam flooding. As estimates of stresses from borehole measurements are improved, it is not uncommon to find that the regional tectonic stress model involving large global averages is significantly different than the local stresses around a borehole that affects the reservior producibility and near-wellbore stability.
The formation stress state is characterized by the magnitude and direction of the three principal stresses. Generally, the overburden stress yields the principal stress in the vertical direction. The overburden stress (Sv) is reliably obtained by integrating the formation mass density from the surface to the dept of interest. Consequently, estimating the other two principal stresses (SHmax and Shmin) in the horizontal plane is the remaining task necessary to fully characterize the formation stress state. FIG. 1 shows various applications of the formation principal stresses together with the well bore (Pp) pressures in well planning, well bore stability and reservoir management.
The far-field formation stresses can be expressed in terms of three principal stresses—they are referred to as the overburden stress Sv, maximum horizontal stress SHmax, and minimum horizontal stress Shmin as shown in FIG. 2. It is known that the wellbore mud overpressure has rather small effect on borehole flexural velocities at low frequencies. In contrast, flexural velocities at low frequencies are more sensitive to changes in the far-field formation horizontal stresses.
Sonic velocities in formations change as a function of rock lithology/mineralogy, porosity, clay content, fluid saturation, stresses, and temperature. To estimate changes in the formation stress magnitudes from measured changes in sonic velocities, it is necessary to select a depth interval with a reasonably uniform lithology, clay content, saturation, and temperature so that the measured changes in velocities can be related to corresponding changes in formation stress magnitudes. Any change in porosity in the chosen depth interval is accounted for by corresponding changes in the formation effective bulk density and stiffnesses. Assuming that the measured changes in sonic velocities are largely caused by changes in stress magnitudes, it is possible to invert borehole sonic velocities for the estimation of changes in formation stress magnitudes (see U.S. Pat. No. 6,351,991). These techniques neither require the presence of a wellbore failure in the form of a breakout or fractures, nor the in-situ rock tensile and compressive strengths. The technique is based on changes in velocities of multiple sonic waves caused by changes in prestress in the propagating medium. It is important to ascertain that one uses sonic velocities outside any mechanically altered annulus caused by borehole-induced stress concentrations and plastic deformations that might also occur.
The propagation of small amplitude waves in homogeneous and anitsotropic materials is described by the linear equations of motion. However, when the material is prestressed, the propagation of such waves are properly described by equations of motion for small dynamic fields superposed on a static bias. A static bias represents any statically deformed state of the medium caused by an externally applied load or residual stresses. The equations of nonlinear elasticity (see B. K. Sinha, “Elastic Waves In Crystals Under A Bias”, Ferroelectrics, (41), 61–73, 1982; A. N. Norris, B. K. Sinha, and S. Kostek, “Acoustoelasticity Of Solid/Fluid Composite Systems”, Geophys. J. Int., (118), 439–446, 1994). The linear equations of motion for isotropic materials contain two independent stiffnesses that are defined by the dynamic Young's modulus (Y) and Poisson's ratio (v) or equivalently, by the two Lame parameters, (λ and μ). The equations of motion or prestressed isotropic materials contain three additional nonlinear stiffness constants (c111, c144, and c155).